inci.matAS {pcds} | R Documentation |
Incidence matrix for Arc Slice Proximity Catch Digraphs (AS-PCDs) - multiple triangle case
Description
Returns the incidence matrix for the AS-PCD
whose vertices are a given 2D numerical data set, Xp
,
in the convex hull of Yp
which is partitioned
by the Delaunay triangles based on Yp
points.
AS proximity regions are defined with respect to
the Delaunay triangles based on Yp
points and vertex
regions are based on the center M="CC"
for circumcenter of each Delaunay triangle
or M=(\alpha,\beta,\gamma)
in barycentric coordinates in the
interior of each Delaunay triangle;
default is M="CC"
i.e., circumcenter of each triangle.
Each Delaunay triangle is first converted to
an (nonscaled) basic triangle so that M
will be the same
type of center for each Delaunay triangle
(this conversion is not necessary when M
is CM
).
Convex hull of Yp
is partitioned
by the Delaunay triangles based on Yp
points
(i.e., multiple triangles are the set of these Delaunay triangles
whose union constitutes the
convex hull of Yp
points).
For the incidence matrix loops are allowed,
so the diagonal entries are all equal to 1.
See (Ceyhan (2005, 2010)) for more on AS-PCDs. Also see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.
Usage
inci.matAS(Xp, Yp, M = "CC")
Arguments
Xp |
A set of 2D points which constitute the vertices of the AS-PCD. |
Yp |
A set of 2D points which constitute the vertices of the Delaunay triangles. |
M |
The center of the triangle.
|
Value
Incidence matrix for the AS-PCD whose vertices are the 2D data set, Xp
,
and AS proximity regions are defined in the Delaunay triangles based on Yp
points.
Author(s)
Elvan Ceyhan
References
Ceyhan E (2005).
An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications.
Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.
Ceyhan E (2010).
“Extension of One-Dimensional Proximity Regions to Higher Dimensions.”
Computational Geometry: Theory and Applications, 43(9), 721-748.
Ceyhan E (2012).
“An investigation of new graph invariants related to the domination number of random proximity catch digraphs.”
Methodology and Computing in Applied Probability, 14(2), 299-334.
Okabe A, Boots B, Sugihara K, Chiu SN (2000).
Spatial Tessellations: Concepts and Applications of Voronoi Diagrams.
Wiley, New York.
Sinclair D (2016).
“S-hull: a fast radial sweep-hull routine for Delaunay triangulation.”
1604.01428.
See Also
inci.matAStri
, inci.matPE
, and inci.matCS
Examples
#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-5; #try also nx<-40; ny<-10 or nx<-1000; ny<-10;
set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))
M<-"CC" #try also M<-c(1,1,1)
IM<-inci.matAS(Xp,Yp,M)
IM
dom.num.greedy(IM) #try also dom.num.exact(IM) #this might take a long time for large nx
IM<-inci.matAS(Xp,Yp[1:3,],M)
inci.matAS(Xp,rbind(Yp,Yp))